Abstract
It was proved by Urbanski and Zdunik (Fund Math 220:23–69, 2013) that for every holomorphic endomorphism \(f:{{\mathbb { P}}}^k\rightarrow {{\mathbb { P}}}^k\) of a complex projective space \({{\mathbb { P}}}^k,k\ge 1\), there exists a positive number \(\kappa _f>0\) such that if \(J\) is the Julia set of \(f\) (i.e. the support of the maximal entropy measure) and \(\phi :J\rightarrow {\mathbb R}\) is a Holder continuous function with \(\sup (\phi )-\inf (\phi )<\kappa _f\) (pressure gap), then \(\phi \) admits a unique equilibrium state \(\mu _\phi \) on \(J\). In this paper we prove that the dynamical system (\(f,\mu _\phi \)) enjoys exponential decay of correlations of Holder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. We also show that the topological pressure function \(t\mapsto P(t\phi )\) is real-analytic throughout the open set of all parameters \(t\) for which the potentials \(t\phi \) have pressure gaps.
Highlights
Let f : Pk → Pk be a holomorphic endomorphism of degree d ≥ 2 of the complex projective space Pk
Denote by J = J ( f ) the Julia set of the map f : Pk → Pk, i.e. the topological support of the measure of maximal entropy
The map f : Pk → Pk is called regular if its exceptional set E = E( f ) does not intersect the Julia set J = J ( f )
Summary
Corollary 2.7 There exists a constant Cl such that, for all n, for every good component V of the set f −n(B ), i.e. V ∈ Gn, and for all x, y ∈ V , we have that exp exp ( Sn φ (x )) ( Sn φ ( y )). Proof This follows directly from the expanding property along good branches (see item (b) in Lemma 2.2) and Hölder continuity of φ
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